wilcox.exact: Wilcoxon Rank Sum and Signed Rank Tests

Description

Performs one and two sample Wilcoxon tests on vectors of data for possibly tied observations.

Usage

## S3 method for class 'default':
wilcox.exact(x, y = NULL, alternative = c("two.sided", "less", "greater"),
            mu = 0, paired = FALSE, exact = NULL,  
            conf.int = FALSE, conf.level = 0.95, ...)
## S3 method for class 'formula':
wilcox.exact(formula, data, subset, na.action, \dots)

Arguments

numeric vector of data values.

an optional numeric vector of data values.

alternative

the alternative hypothesis must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.

a number specifying an optional location parameter.

paired

a logical indicating whether you want a paired test.

exact

a logical indicating whether an exact p-value should be computed.

conf.int

a logical indicating whether a confidence interval should be computed.

conf.level

confidence level of the interval.

formula

a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs a factor with two levels giving the corresponding groups.

data

an optional data frame containing the variables in the model formula.

subset

an optional vector specifying a subset of observations to be used.

na.action

a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

...

further arguments to be passed to or from methods.

Value

A list with class "htest" containing the following components:
statisticthe value of the test statistic with a name describing it.
p.valuethe p-value for the test.
pointprobthis gives the probability of observing the test statistic itself (called point-prob).
null.valuethe location parameter mu.
alternativea character string describing the alternative hypothesis.
methodthe type of test applied.
data.namea character string giving the names of the data.
conf.inta confidence interval for the location parameter. (Only present if argument conf.int = TRUE.)

Details

This version computes exact conditional (on the data) p-values and quantiles using the Streitberg-R"ohmel algorithm for both tied and untied samples.

If only x is given, or if both x and y are given and paired is TRUE, a Wilcoxon signed rank test of the null that the median of x (in the one sample case) or of x-y (in the paired two sample case) equals mu is performed.

Otherwise, if both x and y are given and paired is FALSE, a Wilcoxon rank sum test (equivalent to the Mann-Whitney test) is carried out. In this case, the null hypothesis is that the location of the distributions of x and y differ by mu.

By default (if exact is not specified), an exact p-value is computed if the samples contain less than 50 finite values and there are no ties. Otherwise, a normal approximation is used.

Optionally (if argument conf.int is true), a nonparametric confidence interval for the median (one-sample case) or for the difference of the location parameters x-y is computed. If exact p-values are available, an exact confidence interval is obtained by the algorithm described in Bauer (1972). Otherwise, an asymptotic confidence interval is returned.

References

Myles Hollander & Douglas A. Wolfe (1973). Nonparametric statistical inference. New York: John Wiley & Sons. Pages 27--33 (one-sample), 68--75 (two-sample).

David F. Bauer (1972). Constructing confidence sets using rank statistics. Journal of the American Statistical Association 67, 687--690.

Cyrus R. Mehta and Nitin R. Patel (1998). StatXact-4 for Windows. Manual, Cytel Software Cooperation, Cambridge, USA

Examples

Run this code

## One-sample test.
## Hollander & Wolfe (1973), 29f.
## Hamilton depression scale factor measurements in 9 patients with
##  mixed anxiety and depression, taken at the first (x) and second
##  (y) visit after initiation of a therapy (administration of a
##  tranquilizer).
x <- c(1.83,  0.50,  1.62,  2.48, 1.68, 1.88, 1.55, 3.06, 1.30)
y <- c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29)
wilcox.exact(x, y, paired = TRUE, alternative = "greater")
wilcox.exact(y - x, alternative = "less")    # The same.

<testonly>wt <- wilcox.test(y-x)
  we <- wilcox.exact(y -x)
  wt
  we
  stopifnot(wt$p.value == we$p.value)</testonly>

## Two-sample test.
## Hollander & Wolfe (1973), 69f.
## Permeability constants of the human chorioamnion (a placental
##  membrane) at term (x) and between 12 to 26 weeks gestational
##  age (y).  The alternative of interest is greater permeability
##  of the human chorioamnion for the term pregnancy.
x <- c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46)
y <- c(1.15, 0.88, 0.90, 0.74, 1.21)
we <- wilcox.exact(x, y, alternative = "g")        # greater
wt <- wilcox.exact(x, y, alternative = "g")
<testonly>stopifnot(we$p.value == wt$p.value)
 stopifnot(all(we$conf.int == wt$conf.int))</testonly>

x <- rnorm(10)
y <- rnorm(10, 2)
wilcox.exact(x, y, conf.int = TRUE)

## Formula interface.
data(airquality)
boxplot(Ozone ~ Month, data = airquality)
wilcox.exact(Ozone ~ Month, data = airquality,
            subset = Month %in% c(5, 8))


<testonly>if (!any(duplicated(c(x,y)))) {
    we <- wilcox.exact(x, y, conf.int = TRUE)
    print(we)
    wt <- wilcox.test(x, y, conf.int = TRUE)
    print(wt)
    we$pointprob <- NULL
    we$method <- NULL
    wt$parameter <- NULL
    wt$method <- NULL
    stopifnot(all.equal(wt, we))
    we <- wilcox.exact(x, conf.int = TRUE)
    print(we)
    wt <- wilcox.test(x, conf.int = TRUE)
    print(wt)
    we$pointprob <- NULL
    we$method <- NULL
    wt$parameter <- NULL
    wt$method <- NULL
    stopifnot(all.equal(wt, we))
  }</testonly>
  

# Data from the StatXact-4 manual, page 221, diastolic blood pressure

treat <- c(94, 108, 110, 90)
contr <- c(80, 94, 85, 90, 90, 90, 108, 94, 78, 105, 88)

# StatXact 4 for Windows: p.value = 0.0989, point prob = 0.019

we <- wilcox.exact(contr, treat, conf.int=TRUE)
we
<testonly>stopifnot(round(we$p.value,4) == 0.0989)</testonly>

we <- wilcox.exact(contr, treat, conf.int=TRUE, exact=FALSE)
we
<testonly>stopifnot(round(we$p.value,4) == 0.0853)</testonly>

<testonly># StatXact page 221
  we <- wilcox.exact(treat, contr, conf.int=TRUE)
  stopifnot(we$conf.int[1] == -4)
  stopifnot(we$conf.int[2] == 22)
  stopifnot(we$conf.estimate == 9.5)</testonly> 

# StatXact 4 for Windows: p.value = 0.0542, point prob = 0.019
 
we <- wilcox.exact(contr, treat, alternative="less", conf.int=TRUE) 
we
<testonly>stopifnot(round(we$p.value,4) == 0.0542)</testonly>

# paired observations
# Data from the StatXact-4 manual, page 167, serum antigen level

# StatXact 4 for Windows: p.value=0.0021 (page 168)

pre <- c(149, 0, 0, 259, 106, 255, 0, 52, 340, 65, 180, 0, 84, 89, 212, 554,
500, 424, 112, 2600)
post <- c(0, 51, 0, 385, 0, 235, 0, 0, 48, 65, 77, 0, 0, 0, 53, 150, 0, 165,
98, 0)

we <- wilcox.exact(pre, post, paired=TRUE, conf.int=TRUE)
we
<testonly>stopifnot(round(we$p.value,4) == 0.0021)</testonly>

<testonly># StatXact page 175
  we <- wilcox.exact(post, pre, paired=TRUE, conf.int=TRUE)
  stopifnot(we$estimate > we$conf.int[1] & we$estimate < we$conf.int[2])
  stopifnot(we$conf.int[1] == -292)
  stopifnot(we$conf.int[2] == -54)
  stopifnot(round(we$estimate,1) == -137.8)</testonly>

we <- wilcox.exact(pre,post, paired=TRUE, conf.int=TRUE, exact=FALSE)
we
<testonly>stopifnot(round(we$p.value,4) == 0.0038)</testonly>


<testonly># Hollander & Wolfe (1999), second edition, Example 4.2., page 112

contr <- c(1042, 1617, 1180, 973, 1552, 1251, 1151, 728, 1079, 951, 1319)
SST <- c(874, 389, 612, 798, 1152, 893, 541, 741, 1064, 862, 213)

wilcox.exact(contr, SST, conf.int=TRUE) 

# page 110, Example 4.1

term <- c(0.8, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46)
weeks <- c(1.15, 0.88, 0.90, 0.74, 1.21)

wilcox.exact(weeks, term, conf.int=TRUE)</testonly>

# Hollander & Wolfe, p. 39, results p. 40 and p. 53

x <- c(1.83, 0.50, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.30)
y <- c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29)

we <- wilcox.exact(y,x, paired=TRUE, conf.int=TRUE)
we 

<testonly>stopifnot(round(we$p.value,4) == 0.0391)
  stopifnot(round(we$conf.int,3) == c(-0.786, -0.010))
  stopifnot(round(we$estimate,3) == -0.46)</testonly>

# Hollander & Wolfe, p. 110, results p. 111 and p. 126

x <- c(0.8, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46)
y <- c(1.15, 0.88, 0.90, 0.74, 1.21)

we <- wilcox.exact(y,x, conf.int=TRUE)
we
<testonly>stopifnot(round(we$p.value,4) == 0.2544)
  stopifnot(round(we$conf.int,3) == c(-0.76, 0.15))
  stopifnot(round(we$estimate,3) == -0.305)</testonly>

wel <- wilcox.exact(y,x, conf.int=TRUE, alternative="less")
weg <- wilcox.exact(y,x, conf.int=TRUE, alternative="greater")

<testonly>stopifnot(we$estimate == wel$estimate & we$estimate == weg$estimate)
  stopifnot(we$conf.int[1] <= weg$conf.int[1] & we$conf.int[2] >= wel$conf.int[2])</testonly>

<testonly>stopifnot(wilcox.exact(1:8)$p.value == 0.0078125)
stopifnot(wilcox.exact(c(1:7,7))$p.value == 0.0078125)
stopifnot(wilcox.exact(c(1,1,1))$p.value == 0.25)

x <- rnorm(10)
y <- rnorm(10)
stopifnot(wilcox.test(x,y,conf.int=TRUE)$estimate ==
          wilcox.exact(x,y,conf.int=TRUE)$estimate)
stopifnot(wilcox.test(x,conf.int=TRUE)$estimate ==
          wilcox.exact(x,conf.int=TRUE)$estimate)</testonly>

Run the code above in your browser using DataLab